8 TOPOLOG1CAL GROUPS 35 To complete the proof of (12.8.2.2), it remains to be shown that if H is discrete it is of the form aZ. We may assume that H ^ {0}. Since H = -H, the intersection H n ]0, + oo[ is not empty. If b > 0 belongs to H, the inter- section H n [0, b~] is compact and discrete and therefore^//*? (3.16.3). Let a be the smallest element >0 in this set, and for any x E H let m = [;c/a] be the integral part of x/a: Then x - ma e H, and 0 ^ x — ma < a. This implies that x — ma = 0 and so we have H = aZ. (12.8.3) Let G be a topological group. (i) Let aeG. IfV runs through a fundamental system of neighborhoods of the neutral element e ofG, the sets aV (resp. V a) form a fundamental system of neighborhoods of a. (ii) For each neighborhood U ofe, there exists a neighborhood V ofe such (iii) For each neighborhood U ofe and each aeG, there exists a neighbor- hood Wofe such that aWa""1 c U. (iv) G is Hausdorff if and only if the set {e} is closed in G. The assertion (i) follows from the fact that translations are homeomor- phisms; (ii) expresses that the mapping (x, yj^xy"1 is continuous at (e, e), having regard to the definition of open sets in G x G (12.5); (iii) expresses the continuity of the mapping jci— >axa~*- at the point e. As to (iv), it is clear that if G is Hausdorff the set {e} is closed (12.3.4). Conversely, if {e} is closed and if x, y are distinct points of G, then there exists a neighborhood V of e such that e $ x~ly\, i.e., such that x $yV. If W is a neighborhood of e such that WW""1 c V (which is possible by (ii)), then we have xW n /W = 0, because the relation xw' = yw" with w' and w" in W would imply x = JH/V-* e/WW1 <=:yV. Hence G is Hausdorff. A neighborhood V of e is said to be symmetric if V"1 = V. The symmetric neighborhoods of e form a fundamental system of neighborhoods of e in G, because if U is any neighborhood of e, then so is U""1 (12.8.2) and therefore U n U""1 is a symmetric neighborhood of e contained in U. For each integer n > 0 and each subset V of G we define V" inductively by the rule V" = V""1 • V = V • V"""1. (The set V is not the set of x" with x e V.) If V is a neighborhood of e, then so is V" ^> V for all n ^ 1, and it follows from (12.8J(ii)) that if U is any neighborhood of e and n is an integer > 1, there exists a symmetric neighborhood V of e such that V" c U.point u of ID belonging to the open